A fringe-image phase analysis technique has been used in many fields. A grating pattern is projected onto the surface of an object to be measured and the phase of a grating image which is distorted depending on the height of the object captured by a camera is analyzed to measure a three-dimensional shape and deformation with high accuracy. A technique is known which measures a very small difference in the optical characteristics, the thickness of a transparent material, refractive index distribution, or an inclination angle of an optical component from the analysis of interference fringes by various types of interferometers using a laser beam obtained by a light interference phenomenon. In addition, a technique is known which analyzes the electromagnetic field from the fringe image obtained by electron beam holography. In the medical field, it is necessary to non-invasively measure the tissue quality (the stereoscopic image of a tissue) of a cell which is a product in tissue engineering. In this case, for example, a phase-shift laser microscope developed by Junji Endo at FK OPT LABO CO., LTD. is used. It is very important to provide an analysis method and an analysis device which can rapidly analyze phase information from one or a plurality of phase-shifted fringe images with high accuracy.
It is necessary to extract the phase information of fringes with high accuracy in order to quantitatively calculate the physical amount (for example, a shape, deformation, distortion, or refractive index) of the object to be measured. For example, a Fourier transform (FFT) method, a wavelet method, or a phase shifting method is used as a method for extracting phase information from a fringe image in the related art. The phase analysis methods are classified into a “temporal phase analysis method” which analyzes the phase of the fringe image using temporal intensity information and a “spatial phase analysis method” which analyzes the phase of the fringe image using spatial intensity information. The spatial analysis method can calculate a phase distribution from one fringe grating image and is suitable for dynamic measurement. In contrast, the temporal analysis method can calculate a phase for each pixel of the camera and is suitable for high-resolution analysis.
A phase shifting method has been proposed as one temporal analysis method (Non-patent Document 1). The phase shifting method calculates a phase distribution from T-step digital image data items (hereinafter, a captured digital image with a grating pattern is referred to as a “fringe image”) with an intensity distribution I (x, y; t) represented by the following expression.
                                              ⁢                  Expression          ⁢                                          ⁢          1                                                                                                                        I                ⁡                                  (                                      x                    ,                                          y                      ;                      t                                                        )                                            =                            ⁢                                                                                          I                      a                                        ⁡                                          (                                              x                        ,                        y                                            )                                                        ⁢                  cos                  ⁢                                      {                                                                  2                        ⁢                        π                        ⁢                                                  x                          P                                                                    +                                                                        φ                          0                                                ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    +                                              2                        ⁢                        π                        ⁢                                                  t                          T                                                                                      }                                                  +                                                      I                    b                                    ⁡                                      (                                          x                      ,                      y                                        )                                                                                                                                          =                                ⁢                                                                                                    I                        a                                            ⁡                                              (                                                  x                          ,                          y                                                )                                                              ⁢                    cos                    ⁢                                          {                                                                        φ                          ⁡                                                      (                                                          x                              ,                              y                                                        )                                                                          +                                                  2                          ⁢                          π                          ⁢                                                      t                            T                                                                                              }                                                        +                                                            I                      b                                        ⁡                                          (                                              x                        ,                        y                                            )                                                                                  ,                                                          ⁢                              (                                  t                  =                                      0                    ∼                                          T                      -                      1                                                                      )                                                                        (        1        )            
Here, Ia and Ib indicate the intensity of amplitude (an amplitude component with a frequency 1) and the intensity of a background (an amplitude component with a frequency 0) of the fringe grating, respectively. In addition, P indicates the pitch of the fringe grating, φ0 indicates the initial phase of the fringe grating, and φ indicates the phase value of the fringe image to be finally calculated. Furthermore, x and y indicate position coordinates (in general, integers) on an optical digital camera (the term “optical digital camera” means a digital camera or a video camera which can capture digital image data, regardless of the type of imaging element, such as a CCD sensor or a CMOS sensor and is hereinafter referred to as a “camera”). In addition, t indicates the serial numbers of a plurality of grating images and 2πt/T is a term indicating a phase-shift. In Expression (1), discrete Fourier transform (DFT) is applied to “t” to calculate the angle of deviation of the component with the frequency 1. In this way, the phase distribution is obtained.
                    Expression        ⁢                                  ⁢        2                                                                      φ          ⁡                      (                          x              ,              y                        )                          =                              -            arctan                    ⁢                                                    ∑                                  t                  =                  0                                                  T                  -                  1                                            ⁢                                                          ⁢                                                I                  ⁡                                      (                                          x                      ,                                              y                        ;                        t                                                              )                                                  ⁢                                  sin                  ⁡                                      (                                          2                      ⁢                      π                      ⁢                                                                                          ⁢                                              t                        /                        T                                                              )                                                                                                      ∑                                  t                  =                  0                                                  T                  -                  1                                            ⁢                                                          ⁢                                                I                  ⁡                                      (                                          x                      ,                                              y                        ;                        t                                                              )                                                  ⁢                                  cos                  ⁡                                      (                                          2                      ⁢                      π                      ⁢                                                                                          ⁢                                              t                        /                        T                                                              )                                                                                                          (        2        )            
A grating projection method or a method for measuring the phase of a fringe image using an interferometer generates T-step phase-shifted fringe grating patterns, captures the T-step phase-shifted fringe grating patterns using an optical camera to obtain a plurality of fringe grating images, and analyzes the plurality of fringe grating images using Expression (2). The intensity of the amplitude Ia and the intensity of the background Ib of the fringe grating can be calculated by Expression (3) and Expression (4).
                                              ⁢                  Expression          ⁢                                          ⁢          3                                                                                          I            a                    ⁡                      (                          x              ,              y                        )                          =                              2            N                    ⁢                                                                      [                                                            ∑                                              t                        =                        0                                                                    T                        -                        1                                                              ⁢                                                                                  ⁢                                                                  I                        ⁡                                                  (                                                      x                            ,                                                          y                              ;                              t                                                                                )                                                                    ⁢                      cos                      ⁢                                                                        2                          ⁢                          π                          ⁢                                                                                                          ⁢                          t                                                T                                                                              ]                                2                            +                                                [                                                            ∑                                              t                        =                        0                                                                    T                        -                        1                                                              ⁢                                                                                  ⁢                                                                  I                        ⁡                                                  (                                                      x                            ,                                                          y                              ;                              t                                                                                )                                                                    ⁢                      sin                      ⁢                                                                        2                          ⁢                          π                          ⁢                                                                                                          ⁢                          t                                                T                                                                              ]                                2                                                                        (        3        )                                                          ⁢                  Expression          ⁢                                          ⁢          4                                                                                              ⁢                                            I              b                        ⁡                          (                              x                ,                y                            )                                =                                    1              N                        ⁡                          [                                                ∑                                      t                    =                    0                                                        T                    -                    1                                                  ⁢                                                                  ⁢                                  I                  ⁡                                      (                                          x                      ,                                              y                        ;                        t                                                              )                                                              ]                                                          (        4        )            
In contrast, in the spatial analysis method according to the related art, a sampling moiré method has been proposed (Patent Document 1). The sampling moiré method calculates a phase distribution from a plurality of phase-shifted moiré fringes which are obtained by down-sampling (thinning out) one fringe grating image at an interval close to the pitch of the original grating. FIG. 1 shows the thinning-out process and the intensity interpolation process which are used in the sampling moiré method disclosed in Patent Document 1. Here, the “thinning-out process” extracts intensity data for every M pixel which is arranged at a predetermined interval from the left end or the right end of one fringe grating image (FIG. 1(a)) recorded on the camera. As shown in FIG. 1(b), a plurality of starting points of thinning-out can be changed to obtain a plurality of thinned-off images from one image. In addition, the “intensity interpolation” process interpolates some omitted intensity data using peripheral intensity data, as shown in FIG. 1(c).
FIG. 2 shows the principle of one-shot fringe grating image phase analysis by a one-dimensional sampling moiré method according to the related art. When an optical camera captures the image of an object with a regular grating pattern (FIG. 2(a)), one fringe grating image is obtained. In particular, when a change in the intensity of the grating pattern is a sine wave or a cosine wave, it is represented by Expression (5).
                    Expression        ⁢                                  ⁢        5                                                                      I          ⁡                      (                          x              ,              y                        )                          =                                                            I                a                            ⁢              cos              ⁢                              {                                                      2                    ⁢                    π                    ⁢                                          x                      P                                                        +                                                            φ                      0                                        ⁡                                          (                                              x                        ,                        y                                            )                                                                      }                                      +                          I              b                                =                                                    I                a                            ⁢              cos              ⁢                              {                                  φ                  ⁡                                      (                                          x                      ,                      y                                        )                                                  }                                      +                          I              b                                                          (        5        )            
Here, x and y indicate position coordinates (in general, integers) on the camera and Ia and Ib indicate the intensity of amplitude (an amplitude component with a frequency 1) and the intensity of a background (an amplitude component with a frequency 0) of a fringe grating, respectively. In addition, φ0 indicates the initial phase of the fringe grating and φ indicates the phase value of the fringe image to be finally calculated. Furthermore, P indicates a pitch on the captured image. When an image thinning-out process is performed on the captured one fringe grating image at a pitch M (M is generally an integer) close to P and intensity interpolation is performed using the intensity values of adjacent images, it is possible to obtain a fringe image (hereinafter, referred to as a “moiré fringe image”) with a low spatial frequency, that is, a large pitch. In addition, when the intensity interpolation is performed while changing a starting point m of thinning-out one pixel-by-one pixel, M-step phase-shifted moiré fringe images are obtained, as shown in FIG. 2(b), and can be represented by Expression (6).
                    Expression        ⁢                                  ⁢        6                                                                                                                                  I                  moire                                ⁡                                  (                                      x                    ,                                          y                      ;                      m                                                        )                                            =                            ⁢                                                                    I                    a                                    ⁢                  cos                  ⁢                                      {                                                                  2                        ⁢                                                  π                          ⁡                                                      (                                                                                          1                                P                                                            -                                                              1                                M                                                                                      )                                                                          ⁢                        x                                            +                                                                        φ                          0                                                ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    +                                              2                        ⁢                        π                        ⁢                                                  m                          M                                                                                      }                                                  +                                  I                  b                                                                                                        =                            ⁢                                                                    I                    o                                    ⁢                  cos                  ⁢                                      {                                                                                            φ                          moire                                                ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    +                                              2                        ⁢                        π                        ⁢                                                  m                          M                                                                                      }                                                  +                                  I                  b                                                                                        (        6        )            
The phase of the moiré fringe is shifted from the starting point m of thinning-out by 2π/M. When one-dimensional discrete Fourier transform (DFT) is applied to “m” in Expression (6), it is possible to calculate the phase distribution φmoiré(x, y) of the moiré fringe, as shown in FIG. 2(c).
                    Expression        ⁢                                  ⁢        7                                                                                  φ            moire                    ⁡                      (                          x              ,              y                        )                          =                              -            arctan                    ⁢                                                    ∑                                  t                  =                  0                                                  T                  -                  1                                            ⁢                                                          ⁢                                                                    I                    moire                                    ⁡                                      (                                          x                      ,                                              y                        ;                        m                                                              )                                                  ⁢                                  sin                  ⁡                                      (                                          2                      ⁢                      π                      ⁢                                                                                          ⁢                                              m                        /                        M                                                              )                                                                                                      ∑                                  t                  =                  0                                                  T                  -                  1                                            ⁢                                                          ⁢                                                                    I                    moire                                    ⁡                                      (                                          x                      ,                                              y                        ;                        m                                                              )                                                  ⁢                                  cos                  ⁡                                      (                                          2                      ⁢                      π                      ⁢                                                                                          ⁢                                              m                        /                        M                                                              )                                                                                                          (        7        )            
As shown in Expression (8), the phase distribution of the fringe grating (FIG. 2(d)) can be calculated by adding the phase distribution of the sampling point in the thinning-out process to the phase distribution of the moiré fringe.
                    Expression        ⁢                                  ⁢        8                                                                      φ          ⁡                      (                          x              ,              y                        )                          =                                            φ              moire                        ⁡                          (                              x                ,                y                            )                                +                      2            ⁢            π            ⁢                          x              M                                                          (        8        )            
Expression (8) makes it possible to calculate the phase distribution of the fringe grating using one fringe grating image.
In any method according to the related art, the phase is calculated by one-dimensional discrete Fourier transform, only using one-dimensional phase-shifted intensity information, such as space or time.